Cohomology rings and formality properties of nilpotent groups

TL;DR

This paper explores the relationship between partial formality, resonance varieties, and cohomology rings in finitely generated nilpotent groups, providing criteria for k-formality and computing resonance varieties for specific groups.

Contribution

It introduces the concept of partial formality, establishes criteria linking resonance varieties and cohomology ring generation, and computes these for Heisenberg-type groups.

Findings

Resonance varieties are trivial up to degree k for finitely generated nilpotent k-formal groups.

Cohomology rings of k-formal nilpotent groups are generated in degree 1 up to degree k+1.

Computed resonance varieties for Heisenberg-type groups and determined their degree of partial formality.

Abstract

We introduce partial formality and relate resonance with partial formality properties. For instance, we show that for finitely generated nilpotent groups that are k-formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring of a nilpotent k-formal group is generated in degree 1, up to degree k+1; this criterion is necessary and sufficient for 2-step nilpotent groups to be k-formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.